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Chapter 5: Q. 55 (page 496)

$\int_{0}^{\infty} x \ln x d x$

Short Answer

Expert verified

Undefined.

The limit does not converge.

Step by step solution

01

Given

$\int_{0}^{\infty} x \ln x d x$

02

Calculate integral

Using by parts method of integration.

$\int u d v = u v - \int v d u$

Here, $u = \ln x, d v = x$

$d u = \frac{1}{x}, v = \frac{x^{2}}{2}$

Substitute the above values,

$\int_{0}^{\infty} x \ln x d x = {[\frac{1}{2} x^{2} \ln x - \int \frac{x^{2}}{2} \frac{1}{x} d x]}_{0}^{\infty} = {[\frac{1}{2} x^{2} \ln x - \frac{1}{2} \frac{x^{2}}{2}]}_{0}^{\infty} = {[\frac{1}{2} x^{2} \ln x - \frac{x^{2}}{4}]}_{0}^{\infty} = \infty (Undefined)$

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Most popular questions from this chapter

Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).

Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.

$f (x) = \sec^{- 1} x$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $\int \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$\int \frac{x}{\sqrt{x^{2} - 1}} d x$

See all solutions

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